Large Eddy Simulation, DynamicModel, and Applications
Charles Meneveau
Department of Mechanical Engineering Center for Environmental and Applied Fluid Mechanics
Johns Hopkins University
Mechanical Engineering
Turbulence Summer SchoolMay 2010
Turbulence modeling:
Large-eddy-simulation (LES)
Δ
Δ
G(x): Filter
Large-eddy-simulation (LES) and filtering:
N-S equations:
! !uj
!t+ !uk
! !u j
!xk
= "1#!!p!x j
+ $%2 !u j "!!xk
& jk
Filtered N-S equations:
!u j
!t+!uku j
!xk
= "1#!p!x j
+ $%2uj
! !uj
!t+!uku j"
!xk
= "1#!!p!x j
+ $%2 !u j
where SGS stress tensor is:
! ij = uiu j! " "ui "u j
!uj
!x j= 0
Δ
Δ
G(x): Filter
Energetics (kinetic energy):
! 12 ˜ u j ˜ u j!t
+ ˜ u k! 1
2 ˜ u j ˜ u j!xk
= "!!x j
....( ) " 2# ˜ S jk ˜ S jk " "$ jk˜ S jk( )
!" = # $ jk˜ S jk
Inertial-range flux
Effects of τij upon resolved motions:
E(k)
kLES resolved SGS
!"
!"
! = "u 3
L
!Sij =12
! !ui
!x j
+! !uj
!xi
"
#$%
&'
“SGS energy dissipation”:
!" = # $ jk˜ S jk
If we wish to “control”dissipation of energy we canset τij proportional to -Sij
E.g. Smagorinsky-Lilly model:
! ij
d = "# sgs$ !ui
$x j
+$ !uj
$xi
%
&'(
)*= "2# sgs
!Sij
! sgs = ?? = (velocity " scale) # (length " scale)
! ij
d = "# sgs$ !ui
$x j
+$ !uj
$xi
%
&'(
)*= "2# sgs
!Sij
!sgs = cs"( ) 2 | ˜ S |
Length-scale: ~ Δ (instead of L), Velocity-scale ~ Δ |S|
! sgs ~ "2 | !S |
cs: “Smagorinsky constant”
Theoretical calibration of cs (D.K. Lilly, 1967):
E(k)
kLES resolved SGS
!"
!"
! = "u 3
LE(k) = cK!
2 /3k"5 /3
!" = # = $ % ij!Sij
# = cs2"2 2 | !S | !Sij
!Sij
# & cs2"2 23/2 !Sij
!Sij
3/2
!Sij!Sij =
12
' !ui
'x j
' !ui
'x j
+' !uj
'xi
(
)*+
,-=
=12
[kj2.ii (k) + kik j.ij (k)]d 3k =
|k |</ /"000
12
[k2 ( E(k)4/k2 (1 ii $
k2
k2 )) + 0]d 3k|k |</ /"000
= cK#2 /3 1
2k$5 /3+2 3$1
4/k2 4/k2dk0
/ /"
0 = cK#2 /3 k1/3dk
0
/ /"
0 = cK#2 /3 3
4/"
()*
+,-
4 /3
# & cs2"2 23/2 cK#
2 /3 34
/"
()*
+,-
4 /3(
)*+
,-
3/2
!1 " cs2# 2 3cK
2$%&
'()
3/2
! cs =3cK
2$%&
'()*3/4
# *1
cK = 1.6 ! cs " 0.16
! ij = "2(cs#)2 | !S | !Sij
Theoretical calibration of cs (D.K. Lilly, 1967):
E(k)
kLES resolved SGS
!"
!"
! = "u 3
LE(k) = cK!
2 /3k"5 /3
!" = # = $ % ij!Sij
# = cs2"2 2 | !S | !Sij
!Sij
# & cs2"2 23/2 !Sij
!Sij
3/2
!Sij!Sij =
12
' !ui
'x j
' !ui
'x j
+' !uj
'xi
(
)*+
,-=
=12
[kj2.ii (k) + kik j.ij (k)]d 3k =
|k |</ /"000
12
[k2 ( E(k)4/k2 (1 ii $
k2
k2 )) + 0]d 3k|k |</ /"000
= cK#2 /3 1
2k$5 /3+2 3$1
4/k2 4/k2dk0
/ /"
0 = cK#2 /3 k1/3dk
0
/ /"
0 = cK#2 /3 3
4/"
()*
+,-
4 /3
# & cs2"2 23/2 cK#
2 /3 34
/"
()*
+,-
4 /3(
)*+
,-
3/2
!1 " cs2# 2 3cK
2$%&
'()
3/2
! cs =3cK
2$%&
'()*3/4
# *1
cK = 1.6 ! cs " 0.16
! ij = "2(cs#)2 | !S | !Sij
Theoretical calibration of cs (D.K. Lilly, 1967):
E(k)
kLES resolved SGS
!"
!"
! = "u 3
LE(k) = cK!
2 /3k"5 /3
!" = # = $ % ij!Sij
# = cs2"2 2 | !S | !Sij
!Sij
# & cs2"2 23/2 !Sij
!Sij
3/2
!Sij!Sij =
12
' !ui
'x j
' !ui
'x j
+' !uj
'xi
(
)*+
,-=
=12
[kj2.ii (k) + kik j.ij (k)]d 3k =
|k |</ /"000
12
[k2 ( E(k)4/k2 (1 ii $
k2
k2 )) + 0]d 3k|k |</ /"000
= cK#2 /3 1
2k$5 /3+2 3$1
4/k2 4/k2dk0
/ /"
0 = cK#2 /3 k1/3dk
0
/ /"
0 = cK#2 /3 3
4/"
()*
+,-
4 /3
# & cs2"2 23/2 cK#
2 /3 34
/"
()*
+,-
4 /3(
)*+
,-
3/2
!1 " cs2# 2 3cK
2$%&
'()
3/2
! cs =3cK
2$%&
'()*3/4
# *1
cK = 1.6 ! cs " 0.16
! ij = "2(cs#)2 | !S | !Sij
Theoretical calibration of cs (D.K. Lilly, 1967):
E(k)
kLES resolved SGS
!"
!"
! = "u 3
LE(k) = cK!
2 /3k"5 /3
!" = # = $ % ij!Sij
# = cs2"2 2 | !S | !Sij
!Sij
# & cs2"2 23/2 !Sij
!Sij
3/2
!Sij!Sij =
12
' !ui
'x j
' !ui
'x j
+' !uj
'xi
(
)*+
,-=
=12
[kj2.ii (k) + kik j.ij (k)]d 3k =
|k |</ /"000
12
[k2 ( E(k)4/k2 (1 ii $
k2
k2 )) + 0]d 3k|k |</ /"000
= cK#2 /3 1
2k$5 /3+2 3$1
4/k2 4/k2dk0
/ /"
0 = cK#2 /3 k1/3dk
0
/ /"
0 = cK#2 /3 3
4/"
()*
+,-
4 /3
# & cs2"2 23/2 cK#
2 /3 34
/"
()*
+,-
4 /3(
)*+
,-
3/2
!1 " cs2# 2 3cK
2$%&
'()
3/2
! cs =3cK
2$%&
'()*3/4
# *1
cK = 1.6 ! cs " 0.16
! ij = "2(cs#)2 | !S | !Sij
But in practice (complex flows)
cs = cs (x, t)Ad-hoc tuning?
cs=0.16 works well for isotropic,high Reynolds number turbulence
Examples: Transitional pipe flow: from 0 to 0.16
Near wall damping for wall boundary layers (Piomelli et al 1989)
cs = cs (y)
ycs = 0.16
Measure:
!" = # $ jk˜ S jk
How does cs vary under realistic conditions?Interrogate data:
Measure:
Obtain“empirical” Smagorinsky coefficient = f(x,conditions…):
cs =! " jk
˜ S jk2#2 | ˜ S | ˜ S ij ˜ S ij
$
% & &
'
( ) )
1/ 2
An example result from atmospheric turbulence…:
!"Smag
cs2 = 2"2 | ˜ S | ˜ S ij ˜ S ij
Measure “empirical” Smagorinsky coefficient for atmospheric surface layeras function of height and stability (thermal forcing or damping):
cs =! " jk
˜ S jk2#2 | ˜ S | ˜ S ij ˜ S ij
$
% & &
'
( ) )
1/ 2
Example result: effect of atmosphericstability on coefficient from sonicanemometer measurements inatmospheric surface layer(Kleissl et al., J. Atmos. Sci. 2003)
HATS - 2000(with NCARresearchers:
Horst, Sullivan)Kettleman City
(Central Valley, CA)
Stable stratification
Neu
tral
str
atifi
catio
n
cs = cs (x, t)
How to avoid “tuning” and case-by-caseadjustments of model coefficient in LES?
The Dynamic Model(Germano et al. Physics of Fluids, 1991)
E(k)
k
LES resolved SGS
τ
Germano identity and dynamic model(Germano et al. 1991):
Exact (“rare” in turbulence):
uiuj ! ˜ u i ˜ u j = uiu j ! ˜ u i ˜ u j
E(k)
k
LES resolved SGS
τ
Germano identity and dynamic model(Germano et al. 1991):
Exact (“rare” in turbulence):
uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j
E(k)
k
LES resolved SGS
L τT
Germano identity and dynamic model(Germano et al. 1991):
Exact (“rare” in turbulence):
Tij = ! ij + Lij
uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j
Lij ! (Tij ! " ij ) = 0
E(k)
k
LES resolved SGS
L τT
Germano identity and dynamic model(Germano et al. 1991):
Exact (“rare” in turbulence):
Tij = ! ij + Lij
uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j
!2 cs 2"( )2 | ˜ S | ˜ S ij
!2 cs"( )2 | ˜ S | ˜ S ij
Assumes scale-invariance:
Lij ! (Tij ! " ij ) = 0
where
Lij ! cs2 Mij = 0
Mij = 2!2 | ˜ S | ˜ S ij " 4 | ˜ S | ˜ S ij( )
Germano identity and dynamic model(Germano et al. 1991):
Minimized when:Averaging over regions ofstatistical homogeneityor fluid trajectories
Lij ! cs2 Mij = 0
Over-determined system:solve in “some average sense”(minimize error, Lilly 1992):
! = Lij " cs2 Mij( )2
cs2 =
Lij Mij
Mij Mij
Germano identity and dynamic model(Germano et al. 1991):
Minimized when:Averaging over regions ofstatistical homogeneityor fluid trajectories
Lij ! cs2 Mij = 0
Over-determined system:solve in “some average sense”(minimize error, Lilly 1992):
! = Lij " cs2 Mij( )2
cs2 =
Lij Mij
Mij Mij
Exercise:
(a) Prove the above equation, and
(b) repeat entire formulation for the SGS heat flux vector
modeled using an SGS diffusivity (find Cscalar)
qi = uiT! ! "ui"T
qi = Cscalar!
2 | !S | "!T
"xi
• Mixed tensor Eddy Viscosity Model:• Taylor-series expansion of similarity (Bardina 1980) model (Clark 1980, Liu, Katz & Meneveau (1994), …)• Deconvolution: (Leonard 1997, Geurts et al, Stolz & Adams, Winckelmans etc..)• Significant direct empirical evidence, experiments:
• Liu et al. (JFM 1999, 2-D PIV)• Tao, Katz & CM (J. Fluid Mech. 2002):
tensor alignments from 3-D HPIV data• Higgins, Parlange & CM (Bound Layer Met. 2003):
tensor alignments from ABL data• From DNS: Horiuti 2002, Vreman et al (LES), etc…
( )k
j
k
inlijSij x
uxuCSSCT
!!
!!"+"#=
~~2~~)2(2 22
ijSk
j
k
inl
mnlij SSC
xu
xuC ~~)(2
~~22 !"
##
##!=$
Two-parameter dynamic mixed modeljijiijijij uuuuTL ~~~~ !=!" #
!" ijd
˜ S ij
Similarity, tensor eddy-viscosity, and mixed models
Reality check:
Do simulations with these closures producerealistic statistics of ui(x,t)?
• Need good data• Need good simulations
• Next: Summary of results from Kang et al. (JFM 2003)•Smagorinsky model, •Dynamic Smagorinsky model,•Dynamic 2-parameter mixed model
Remake of Comte-Bellot & Corrsin (1967)decaying isotropic turbulence experiment
at high Reynolds number
Contractions
Active GridM = 6 "
1x2x
M20 M30 M40 M48
Test Section
1u2u X-wire probe array
Δ1 = 0.01mΔ2 = 0.02mΔ3 = 0.04mΔ4 = 0.08m
0.91mFlow
h1 = 0.005mΔ1 = 0.01m h3 = 0.02m
Δ3 = 0.04mΔ
1x2x
M20 M30 M40 M48
1u2u
Flow
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
100 101 102 103 104
E(!)
!
x1/M=20
x1/M=48
Initial condition for LES
(m3s-2)
(m-1)
Res
ults
LESTemporally Decaying Turbulence
3-D Filter
ExperimentSpatially Decaying Grid Turbulence
2-D Box Filter tux 11 =
Taylor Hypothesis
LES of Temporally Decaying Turbulence
Pseudo-spectral code: 1283 nodes, carefully dealiased (3/2N) All parameters are equivalent to those of experiments. Initial energy distribution: 3-D energy spectrum at x1/M = 20
LES Models: standard Smagorinsky-Lilly model,dynamic Smagorinsky and
dynamic mixed tensor eddy-visc. model
Results: Dynamic Model Coefficients
Dynamic Smagorinsky Dynamic Mixed tensor eddy visc:
ijSk
j
k
inl
mnnlij SSC
xu
xuC ~~)(2
~~22 !"
##
##!=$
0
0.05
0.1
0.15
0.2
0 10 20 30 40 50
Cs
x1/M
Cs
0
0.05
0.1
0.15
0.2
0 10 20 30 40 50
Cs
Cnl
x1/M
Cnl
Cs
Cnl
Cs
1/12 from the first orderapproximation of τij
! ijdyn"Smag = "2 Lij M ij
M ij M ij#2 ˜ S ˜ S ij
3-D Energy Spectra (LES vs experiment)
Dynamic Smagorinsky
10-5
10-4
10-3
10-2
10-1
100
100 101 102 103
E(!)
!
x1/M=20
x1/M=48
• cutoff grid filter (Δ = 0.04 m)• cutoff test filter at 2Δ
Exp.
Dynamic Mixed tensor eddy-visc. model
10-5
10-4
10-3
10-2
10-1
100
100 101 102 103
E(!)
!
x1/M=20
x1/M=48
• cutoff grid filter (Δ = 0.04 m)• cutoff test filter at 2Δ
3-D Energy Spectra (LES vs experiment)
PDF of SGS Stress
SGS Stress
0
5
10
15
20
25
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
x1/M=48
Exp.
Dynamic Mixed Nonlinear
Smagorisnky
Dynamic Smagorisnky
2112
~/ rmsu!
Dynamic mixed tensor eddy-visc model predicts PDF of theSGS stress accurately.
212112~~uuuu !"#
(LES vs experiment)
E(k)
k
LES resolved SGS
L τT
Germano identity and dynamic model(Germano et al. 1991):
Exact (“rare” in turbulence):
Tij = ! ij + Lij
uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j
!2 cs 2"( )2 | ˜ S | ˜ S ij
!2 cs"( )2 | ˜ S | ˜ S ij
Assumes scale-invariance:
Lij ! (Tij ! " ij ) = 0
where
Lij ! cs2 Mij = 0
Mij = 2!2 | ˜ S | ˜ S ij " 4 | ˜ S | ˜ S ij( )
Germano identity and dynamic model(Germano et al. 1991):
Minimized when:Averaging over regions ofstatistical homogeneityor fluid trajectories
Lij ! cs2 Mij = 0
Over-determined system:solve in “some average sense”(minimize error, Lilly 1992):
! = Lij " cs2 Mij( )2
cs2 =
Lij Mij
Mij Mij
Problem: what to do for non-homogeneous flowswithout directions over which to average(“learn”, or “assimilate” larger-scale statistics?)
Lagrangian dynamic model (M, Lund & Cabot, JFM 1996):
Average in time, following fluid particles for Galilean invariance:
E = Lij ! Cs2Mij( )2
!"
t
# 1T
e!
(t -t')T dt '
Lagrangian dynamic model (M, Lund & Cabot, JFM 1996):
Average in time, following fluid particles for Galilean invariance:
! E = 0 " Cs2 =
LijMij #$
t
%1T
e#
(t - t' )T dt'
Mij Mij #$
t
%1T
e#
(t - t' )T dt'
E = Lij ! Cs2Mij( )2
!"
t
# 1T
e!
(t -t')T dt '
!MM = Mij Mij "#
t
$ 1T
e"
(t - t' )T dt'
!LM = LijMij "#
t
$ 1T
e"
(t -t')T dt '
Lagrangian dynamic model (M, Lund & Cabot, JFM 1996):
Average in time, following fluid particles for Galilean invariance:
! E = 0 " Cs2 =
LijMij #$
t
%1T
e#
(t - t' )T dt'
Mij Mij #$
t
%1T
e#
(t - t' )T dt'
E = Lij ! Cs2Mij( )2
!"
t
# 1T
e!
(t -t')T dt '
With exponential weight-function, equivalent to relaxation forward equations:
!MM = Mij Mij "#
t
$ 1T
e"
(t - t' )T dt'
!LM = LijMij "#
t
$ 1T
e"
(t -t')T dt '
!"LM
!t+ ˜ u k
!"LM
!xk
=1T
(LijMij # "LM )
!"MM
!t+ ˜ u k
!"MM
!xk
=1T
(Mij Mij # "MM )
cs2 =
!LM (x, t)!MM (x, t)
Lagrangian dynamic model has allowed applying the Germano-identity to a number of complex-geometry engineering problems
LES of flows in internal combustion engines:Haworth & Jansen (2000)Computers & Fluids 29.
LES of structure of impinging jets:Tsubokura et al. (2003)Int Heat Fluid Flow 24.
LES of flow over wavy walls Armenio & Piomelli (2000)Flow, Turb. & Combustion.
Examples:
LES of flow in thrust-reversers Blin, Hadjadi & Vervisch (2002)J. of Turbulence.
Examples:
LES of flow in turbomachinery Zou, Wang, Moin, Mittal. (2007)Journal of Fluid Mechanics.
Examples:
Examples:
LES of convective atmospheric boundary layer: Kumar, M. & Parlange (Water Resources Research, 2006)
• Transport equation for temperature• Boussinesq approximation• Coriolis forcing• Lagrangian dynamic model with assumed β=Cs(2 Δ)/Cs(Δ)• Constant (non-dynamic) SGS Prandtl number Prsgs=0.4• Imposed surface flux of sensible heat on ground• Diurnal cycle: start stably stratified, then heating….
Imposed ground heat flux during day:
• Diurnal cycle: start stably stratified, then heating….
Examples:
Resulting dynamic coefficient (averaged):
Consistent with HATS field measurements:
Large-eddy-Simulation of atmospheric flow over fractal trees:
Downtown Baltimore:
URBAN CONTAMINATIONAND TRANSPORT
Yu-Heng Tseng, C. Meneveau & M. Parlange, 2006 (Env. Sci & Tech. 40, 2653-2662)
Momentum and scalar transport equations solved using LES andLagrangian dynamic subgrid model. Buildings are simulated usingimmersed boundary method.
wind
Useful references on LES and SGS modeling:
• P. Sagaut: “Large Eddy Simulation of Incompressible Flow” (Springer, 3rd ed., 2006)
• U. Piomelli, Progr. Aerospace Sci., 1999
• C. Meneveau & J. Katz, Annu Rev. Fluid Mech. 32, 1-32 (2000)
• C. Meneveau, Scholarpedia 5, 9489 (2010).
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